Propagation characteristics of hydromagnetic waves in various layers of the ionosphere

PImet.

Space Sci.. Vol.

24, pp. 907 to 914. Pergmon

Press,

1976.

Printed

in Northem~ Ireland

PROPAGATION CHARACTERISTICS OF HYDROMAGNETIC WAVES IN VARIOUS LAYERS OF THE IONOSPHERE Department

MANH DUNG DUONG of Physics, University of Newcastle, New South Wales, Australia (Received 23 February 1976)

Abstract-This paper derives the basic propagation characteristics of hydromagnetic waves in various layers of the ionosphere. It is shown that propagation in the upper ionosphere and the Fz layer is largely isotropic. In the lower region of the ionosphere there are two possible modes of propagation, both being anisotropic. Propagation characteristics of waves in this lower region, however, are relatively independent of the direction of horizontal propagation. Calculations of intrinsic wave attenuation show that ducted propagation of PC 1 signals over appreciable horizontal distances may only take place in the upper layers of the ionosphere. 1. INTRODUCIION

It is now generally accepted that the naturally occurring PC 1 micropulsations observed at the ground are the result of hydromagnetic disturbances high in the magnetosphere. The disturbances propagate through the magnetosphere and the ionosphere as hydromagnetic waves. There has been a number of theoretical studies on propagation of hydromagnetic waves in the ionosphere (e.g. Fejer, 1960, Francis and Karplus, 1960; Prince and 1964; Field and Greifinger, 1965; Bostick, Greifinger, 1972). Most of these analyses have only considered special cases of propagation, and quite often the ionospheric models used are complicated and the systems of naming wave modes confusing. This paper presents a generally applicable discussion of hydromagnetic propagation in the ionosphere. It is shown that the basic propagation characteristics of hydromagnetic waves in various ionospheric layers may be obtained using a relatively simple approach. The results of this study also form the basis for the theoretical work on the ducted propagation of hydromagnetic waves in the ionosphere which is presented in a separate paper.

4. Collisions between electrons and ions are negligible. 5. Pressure and gravitational forces are small compared to electrostatic restoring forces. Under these assumptions the ionosphere may be regarded as an anistropic plasma. Propagation of hydromagnetic waves in such a medium is governed by the wave equation: VxVxE-iop

J=O,

J=u,,(E.b)b+al[E-(E.b)b]-q(Exb),

GENERAL

WAVE

EQUATIONS

J=u,E-uZ(Exb) with -x

The theoretical treatment of hydromagnetic waves in the ionosphere is based on the following assumptions: 1. The static terrestrial magnetic field is uniform. 2. The magnetic field of any disturbances to be considered is negligible when compared to the static terrestrial field. 3. The ionosphere is still, flat and inhomogeneous only in the vertical direction. Variations of the medium properties in horizontal directions are neglected.

907

(2)

where b is a unit vector in the direction of the static terrestrial field and CT,,,u1 and u2 are, respectively, longitudinal, Pedersen and Hall conductivities. For altitude above 90 km, and for wave frequencies in the Pcl range, it is often assumed that E -b =0 (Francis and Karplus, 1960; Smiley, 1964; Greifinger and Greifinger, 1968) so that the expression for J reduces to:

vi - io

ni 2.

(I)

where current density J is given by:

02=-

y;+f& vi - iw Ul, a

where vi is the collision frequency of ions with neutral particles, ni is the ion cyclotron frequency and V, is the Alfvkn speed. It is convenient to orient the coordinate system so that the z-axis is vertically downward and the x-axis in the direction of horizontal propagation (Fig. 1). Because the ionosphere is assumed inhomogeneous only in the vertical direction, the electric field intensity may be

908

MANHDUNGDUONG Uniform collisionless plasma

Upper ionosphere

z z

I

23

Fz layer 22 Region Lower ionosphere

---

I

-------------_zo Region2

FIG. 1. RECTANGULAR COORDINATE SYSTEM ROMAGNETICWAVEPROPAGATION.

FOR

HYD..,

:

.’

FIG.

2.

where k is the horizontal wave number. Within the chosen rectangular coordinate system, the general wave equation (1) may be written in the following component form: -. E*+ik;E, - iop (a, E, - uz b,E, + uz b,E,) = 0 kz)Ey

>

- iwp(u~E,

+ u,b,E,

- qb,E,)

(7)

=0

k2E,+ikAEx -iq.4uIEr+u2bxEy-u2byEx)=0

MODEL

OF THE

MODEL OF THE SYSTEM.

,,.

.:

IONOSPHERE

Taking the height profiles of the AlfvCn speed used by Jacobs and Watanabe (1962) as being representative, one may divide the ionosphere into three qualitatively distinct regions. (i) The outermost or upper ionosphere, an essentially collision-free region in which the Alfven speed increases exponentially with altitude from a value V2 at its lower boundary, to a value Vi as one approaches the magnetosphere. (ii) The central region, or Fz layer, in which the Alfven velocity remains close to its minimum value V,. Frequency of collisions between ions and neutral particles is small in this region.

ZG

ground

IONOSPHERE-EARTH

(iii) The lower ionosphere, a combined E and F, region in which the AlfvCn speed also remains at an approximately constant value V,, which is greater than V,. Collision frequency in this region is significantly large and increases with decreasing altitude. Below the E and F, regions is the ionosphereEarth cavity (vacuum). For frequencies in the Pcl range the Earth may be regarded as a perfect conductor. Figure 2 is a schematic diagram of the ionospheric model. The representative values of the relevant parameters for four principal ionospheric conditions are given in Table 1. The model employed here is basically similar to that used by

,

where b,, b, and b, are the x, y and z components of the unit vector b. 3. PHYSICAL

2.:.

Perfectly-conducting

SCHEMATIC

E(x, y, z) = E(r)eik”,

(--$+

.: ,.;..., ”

expressed in the form:

-5

:

TABLE

1. VALUES

OF IONOSPHERICPARAMETERS

909

Ionospheric propagation of hydromagnetic waves Smiley (1973).

(1964)

and

Greifinger

and

Greifinger

I5

I

4. WAVES IN THE UPPER IONOSPHERE

The upper ionosphere, in which the AlfvCn speed is an exponential function of altitude, may be approximated by a large number of thin layers within each of which the AlfvCn speed remains at a constant value. Propagation of waves in each of these layers is governed by the dispersion relation developed in the Appendix. Considerable simplification, however, is possible due to the fact that within the upper ionosphere collisions between neutral and charged particles are insignificant. With vi = 0 and w
Day

maximum

/

a 5 ,. _ “d _ f _ $ 8 8 5-

(8)

where ur = -(iw/pV,*). Upon substituting the expression for u1 into (8) one obtains the following dispersion relation: h== 02/V,,*

FIG. 3. WAVES

(9)

which indicates that, under the conditions assumed, there is only one predominant mode of propagation for Pcl waves in the upper ionospheric layers. This wave mode is in fact the familiar modified Alfven or isotropic fast mode. There is no intrinsic wave attenuation in the upper ionosphere because this region has been considered collision-free.

3.0 2.0 Frequency, Hz

IO

0

INTRINSIC IN THE

PHASE

F2 LAYER

CONSTANT

4.0

OF

HYDROMAGNETIC

AS A FUNGI-ION

OF FREQUENCY.

within the F2 layer. The total wave number h, which appears in equation (lo), is complex. Its real and imaginary parts are given by:

5. WAVES IN THE Fz LAYER

In the F, layer, where the AlfvCn speed has a constant value V,, the frequency of collisions, vi, between ions and neutral particles increases with decreasing altitude. Throughout the F, layer, however, vi remains small compared to Q, the ioncyclotron frequency, so that, as an approximation, it is assigned a mean value v,. Under this assumption F2 region becomes a homogeneous ionospheric layer the propagation characteristics of which may be determined from the dispersion relation obtained in the Appendix. By making use of the inequalities vi <
h’=$+iF. 2

Again

only the isotropic

(10)

2

fast mode is present

The real part of h gives the intrinsic phase constant and the imaginary part the intrinsic attenuation coefficient. Variations of Re(h) and Im(h) with wave frequency for different ionospheric conditions are shown in Figs. 3 and 4. As can be seen in Fig 4, the intrinsic wave attenuation remains close to a constant value for all wave frequencies in the Pcl range. 6. WAVES IN THE LOWER IONOSPHERE

Throughout the lower ionosphere, the Alfven speed is assumed to have a constant value V,, which is much greater than V,. In this region of space the collision frequency Vi increases very rapidly with decreasing altitude. As a first approximation, however, the lower ionosphere is considered as being composed of a number of horizontal strata within each of which the collision frequency assumes a constant value, so that the dispersion relation developed in the Appendix is still applicable. Ideally

MANH Duo

910

DIJONG

2f

E :: 2c B “0’

Day maximum z It .!$ E 8 s ._ : 0 t a

IC

Day minimum !

Night maximum

I I .o

C

hG. 4.

t

3.0 2.0 Frequency, Hz

4.0

I.0

0

ROMAGNETIC

WAVES

ATIENUATION IN THE,

F2

COfiFFICIENT LAYER

OF

As A FUNCTION

HYD-

5.

WAVES

INTRINSIC IN

PHASE

REGION

OF

1 OF

FUNCnON

4.0

3.0

Frequency, FIG.

hTRINSIC

2.0

Hz

CONSTANT THE

OF

LOWER

HYDROMAGNETIC

IONOSPHERE

As

A

OF FREQUENCY.

FREQUENCY.

complex form of equation the number of layers used in the approximation should be large. However, in this study, where the aim is to obtain some general understanding of wave characteristics in the lower ionosphere, only two such layers are used. The first layer is situated above a height of lSOkm, called region 1 of the lower ionosphere, and has a constant collision frequency q,. Below 15Okm is the region 2 of the lower ionosphere in which vi assumes a much greater value vtz. The frequency of collisions, v,,, between neutral and charged particles within region 1 of the lower ionosphere is still small compared to sli, the ion cyclotron frequency. Consequently, the dispersion relation for waves in this layer also has the simplified form: h* = iwv,*, with a, = - (iw/pV,Z)[l+ i(vJo)]. Substituting V, = V, and vi = vL,, one obtains the following dispersion relation:

Propagation of waves in region 1 of the lower ionosphere, therefore, is still mainly isotropic. The intrinsic phase constant and attenuation coefficient in region 1 of the lower ionosphere are shown as functions of wave frequency in Figs. 5 and 6. In region 2 of the lower ionosphere, the inequality vI<
iocwi h2=T-

(A4):

[(2+ ur*)f rJ4(a-

l)+a*r*]

where:

The dispersion relation shown above indicates that there are two modes of propagation, both being dependent on propagation direction. The

2o2 16-

0

I .o

2.0

3.0

Frequency, FIG.

6.

INTRINSIC

ROMAGNETIC

ATTENUATION

WAVES

PHERE

IN REGION

AS A FUNCITGN

4.0

5.0

Hz COEFFICIENT

1 OF THE

OF

LOWER

OF FREQUENCY.

HYDIONOS-

911

Ionospheric propagation of hydromagnetic waves 8=00; JI=O”; f= 1.0 HZ

IO,

-

Attenudion

----Phase

-

Attenuation

----Phase

1 0

I

.o

2.0

CIENT

7. VARIATIONS AND

TIC WAVES IONOSPHERE

OF

INTRINSIC WITH

FREQUENCY

UNDER

4.0

Hz

INTRINSIC

PHASE

ATTENUATION

CONSTANT

-2

I

5.8

COEFFI-

OF HYDROMAGNE-

IN REGION

NIGHT-TIME

constant

constant 3.0

Frequency. FIG.

coefficient

, IO

coefficient

2 OF THE LOWER

MINIMUM

SUNSPOT CONDI-

TION.

FIG.

9.

CIENT

VARIATIONS AND

TIC WAVES

WITH 4IN

UNDER

wave mode that corresponds to the upper sign (+) will be called mode 1, and that which corresponds to the lower sign (-) mode 2. Figures 7-14 show the dependence of propagation characteristics of waves in region 2 of the lower ionosphere on various parameters. Intrinsic wave attenuation in 5-

OF

INTRINSIC

INTRINSIC

PHASE REGION

NIGHT-TIME

ATTENUATION

CONSTANT

2 OF THE LOWER

MINIMUM

COEFFI-

OF HYDROMAGNE-

SUNSPOT

IONOSPHERE

CONDITION.

this region tends to increase more rapidly with increasing wave frequency. Propagation characteristics show complex variations with 0 and 4 but remain fairly constant for all values of $. There is only a slight increase in wave attenuation, in the case of mode 1, and a slight decrease in wave attenuation, in the case of mode 2, as propagation 8.00; IO

--_

.

JI=OO;f=I.OHr IO

7-\..

O-

---Phase

FIG. CIENT

8.

VARIATIONS

AND

TIC WAVES IONOSPHERE

INTRINSIC WITH

40

30 2.0 Frequency, Hz

10

3

OF

INTRINSIC

PHASE

FREQUENCY

UNDER

43 degree ATTRNUATION

CONSTANT IN REGION

DAYTIME TION.

constant

I

5.0”

MAXIMUM

COEFFI-

OF HYDROMAGNE-

2 OF THE LOWER SUNSPOT

CONDI-

FIG. CIENT

10.

VARIATIONS

AND

TIC WAVES UNDER

INTRINSIC

OF INTRINSIC PHASE

WITH 4 IN REGION DAYTIME

A-ITENWATION

CONSTANT

2 OF THE LOWER

MAXIMUM

SUNSPOT

COEFFI-

OF HYDROMAGNEIONOSPHERE

CONDITION.

MANH DUNG DUONG

912

----

Attenuation

-

Phase constant

coefficient

E E t

$8 ‘D

Mode

I

B

Mode 2

0

20

JI,

FIG.

11.

AND

CIENT TIC

VARIATIONS INTRINSIC

WAVES

SPHERE

WITH

UNDER

6

OF INTRINSIC PHASE IN

CONSTANT

REGION

NIGHT-TIME

ATTENUATION

2

OF HYDROMAGNE-

OF

MINIMUM

COEFFI-

THE

LOWER

SUNSPOT

IONO-

8=1V;J/=O”;

10,

‘\

AND

TIC WAVES

IO

.I0

____

-6

FIG.

12.

CIENT TIC

VARIATIONS

AND

WAVES

SPHERE

INTRINSIC WITH

UNDER

60

0

ATTENUATION

CONSTANT

REGION

8.0’;

2

MAXIMUM

OF

N0

Ci x E

THE

LOWER

IONOSPHERE

CONDITION.

f=l.OHz

_______

IO

-6

Mode I

E 5

6-

Attenuation coefficient ----Phase constant

.$ ;; 5 x

COEFFIIONO-

CONDITION.

6

L 0 x $ +

s F s g

4-

a

2-

0

OF HYDROMAGNE-

SUNSPOT

THE LOWER SUNSPOT

//

a-

-4

Mode 2 --w-1____

--__----_______-_____.

0

FIG. CIENT

?? % 2 a

-2

20

40 9,

OF INTRINSIC IN

MINIMUM

I

,

degree

PHASE

DAYTIME

60

2 OF

REGION

COEFFI-

OF HYDROMAGNE-

i

f

0,

+IN

ATTENUATION

CONSTANT

______----------

Attenuation coefficient Phose constant

40

degree

PHASE

NIGHT-TIME

60

though the results obtained in this section are relatively new, they appear to still fit in with the usual theory (Greifinger and Greifinger, 1968; Smiley, 1964) where it is thought that within the lower ionosphere, the fast (or modified Alfven) mode is coupled to the slow (or AlfvCn) mode. This coupling is likely to produce wave modes of a hybrid nature like those revealed by the present study.

e

20

WITH

60

OF INTRINSIC

INTRINSIC

UNDER

E

0

VARIATIONS

+=l5O;

f=l.OHz

-----

‘\

CIENT

CONDITION.

direction deviates from the magnetic meridian plane. In all cases, mode 2 suffers less intrinsic attenuation than mode 1. As can be seen, the predominant collisions between neutral and charged particles in region 2 of the lower ionosphere give rise to wave modes that are much more complex than the propagation modes often found in a cold uniform plasma. Al-

13.

FIG.

1

40

14.

VARIATIONS

AND

TIC WAVES UNDER

INTRINSIC

OF INTRINSIC PHASE

WITH I& IN REGION DAYTIME

60

60

O

degree ATTENUATION

CONSTANT

2 OF

MAXIMUM

THE LOWER

SUNSPOT

COEFFI-

OF HYDROMAGNEIONOSPHERE

CONDITION.

Ionospheric propagation of hydromagnetic waves 7. SUMMARY

OF CONCLUSIONS

By means of a simplified theoretical model it is possible to obtain some basic propagation characteristics of Pcl hydromagnetic waves in the various regions of the ionosphere. In the upper ionosphere and the F, layer, where collision effects are insignificant, hydromagnetic waves in the Pcl frequency range propagate mainly in the isotropic fast mode. Within the lower ionosphere, where collisions between neutral and charged particles are predominant, there are two possible modes of propagation, both being anisotropic. In general, the changes in propagation characteristics as propagation direction deviates from the geomagnetic meridian plane are not significant. The theoretical considerations presented in Sections 4 and 5 support the belief held by many authors that on arriving at the Earth’s ionosphere, hydromagnetic waves propagate horizontally in the isotropic fast mode only. Results of theoretical calculations of intrinsic wave attenuation also indicate that horizontal ducting of Pcl signals may take place in the upper ionosphere and the F2 region. These are the layers that have very little or no intrinsic attenuation, and consequently may sustain ducted propagation of hydromagnetic waves over large distances. In the lower ionosphere, wave attenuation due to medium properties is so large that ducting of waves becomes insignificant.

Acknowledgement-The author would like to thank Dr. B. J. Fraser for his supervision on this work. Funds have been provided by the Australian Research Grants Committee and the Radio Research Board. REFERENCES

Fejer, J. A. (1960). Hydromagnetic wave propagation in the ionosphere. J. atnws. rem Phys. 18,135. Field, E. C. and Greiflnger, C. (1965). Transmission of geomagnetic micronulsations through the ionosphere and lower exosphere. J. geophys. R&. 70, 4885. Francis. W. E. and Karnlus. R. (1960). Hvdromaanetic waves in the ionosphere. j. geobhys. ‘Res.S65, 3553. Greifinger, C. and Greifinger, P. S. (1968). Theory of hydromagnetic propagation in the ionospheric waveguide. J. geoihyi. Res. 73, 7473. Greifinger, P. S. (1972). Ionospheric propagation of oblique hydromagnetic plane waves at micropulsation freauencies. J. aeoDhvs. Res. 77. 2377. Greil&ger, C. a& &&finger, PI S. (1973). Waveguide propagation of micropulsations out of the plane of the geomagnetic meridian. .I. geophys. Res. 78, 4611. Jacobs, J. A., and Watanabe, T. (1962). Propagation of hydromagnetic waves in the lower exosphere and the origin of short period geomagnetic pulsations. .I. armos. ten. Phys. 24, 413. Prince, Jr. C. E. and Bostick, Jr., F. X. (1964). Ionospheric transmission of transversely propagated plane

913

waves at micropulsation frequencies and theoretical power spectrums. J. geophys. Res. 69, 3213. Smiley, R. F. (1964). The penetration of oblique hydromagnetic waves through horizontally stratified plasma and unionized media, with applications to the lower ionosphere. Rept. 13, contract DA-36-039-SC-89177, U.S. Army Electron. Labs. Fort Monmouth, NJ.

APPENDIX

Wave Propagation in a Homogeneous Ionospheric Layer The ionosphere, being assumed inhomogeneous only in the vertical or z-direction, is approximately equivalent to a number of discrete thin horizontal strata, each of which being uniform in all directions. By making the layers indefinitely thin we can approach the actual medium as closely as we please. This appendix derives the dispersion relation for waves propagating in a homogeneous ionospheric stratum. Within each layer, the ionospheric parameters are uniform in all directions. One, therefore, may substitute alar = iy, where y is the vertical wave number, into the wave equation (7) to obtain: y2Ex - kyE, - iop(u,E,

- u2b& + u2bYE,) = 0

Let 0 be the angle formed by the resultant wave. vector II and the horizontal wave vector k, which is in the direction of the x-axis, then: k = h cos 0 and y = h sin 8. Substituting the expressions for k and y into (Al) and rearranging terms give the following system of equations: (h’sin 0-iop.ul)Ex+io~2bzEY -(h2sin8cos8+iw~2bY)EY=0 - iop.u2bzEx +(h’-

iop.u,)E, + iopz b,E, = 0

-(h2sinecose-io~2bY)Ex -i~pu~b~E~ +(h2 cm2 8-io~i)E,

This

W)

=0

In order for (A2) to have non-trivial the following determinant must be zero: hZ sin’ 0 - iwpcl -iot.t+b. -hZsinf3ws0+iopa~by

’

solutions,

-h* sin 8 cm 8 - iopu2by iowzb, hZ - iopz~Lo1 iopmb. h2 cd 0 - iopu, -iopn2bx =o.

yields a quadratic in h*: Ah4+Bh2+C=0

(-43)

where A=-u, B = iop{2ulZ + uz2[(bx sin 0 - b,

cos

8)’

+

bY2D

c = 02~2ul(u12+u22). Equation (A3) is the dispersion waves in a homogeneous ionospheric

relation stratum.

for The

914

MANH DUNG DLJONG where

roots of this equation are given by:

h2= Upon substituting one obtains: fi2=_

i-m 2

o=(bXsin0-bzcos8)2+by2

-BktJB’-4AC

2A

the expressions

Vi- io *II‘4 . for A, B and C

[(2+ar2)*r~4(~-l)+azr2]

(A4)

For altitudes above 150 km, Vi<